This model is a polytomous model proposed by Falk & Cai (2016) and is based on the generalized partial credit model (Muraki, 1992).
rpf.gpcmp(outcomes = 2, q = 0, multidimensional = FALSE)an item model
The number of possible response categories.
a non-negative integer that controls the order of the polynomial (2q+1) with a default of q=0 (1st order polynomial = generalized partial credit model).
whether to use a multidimensional model.
Defaults to FALSE. The multidimensional version is not yet
available.
The GPC-MP replaces the linear predictor part of the generalized partial credit model with a monotonic polynomial, \(m(\theta;\omega,\xi,\mathbf{\alpha},\mathbf{\tau})\). The response function for category k is:
$$\mathrm P(\mathrm{pick}=k|\omega,\xi,\alpha,\tau,\theta) = \frac{\exp(\sum_{v=0}^k (\xi_k + m(\theta;\omega,\xi,\mathbf{\alpha},\mathbf{\tau})))}{\sum_{u=0}^{K-1}\exp(\sum_{v=0}^u (\xi_u + m(\theta;\omega,\xi,\mathbf{\alpha},\mathbf{\tau})))} $$
where \(\mathbf{\alpha}\) and \(\mathbf{\tau}\) are vectors
of length q. The GPC-MP uses the same parameterization for the polynomial
as described for the logistic function of a monotonic polynomial (LMP).
See also (rpf.lmp).
The order of the polynomial is always odd and is controlled by
the user specified non-negative integer, q. The model contains
1+(outcomtes-1)+2*q parameters and are used as input to the rpf.prob
function in the following order:
\(\omega\) - natural log of the slope of the item model when q=0,
\(\xi\) - a (outcomes-1)-length vector of intercept parameters,
\(\alpha\) and \(\tau\) - two parameters that control bends in
the polynomial. These latter parameters are repeated in the same order for
models with q>0. For example, a q=2 polynomial with 3 categories will have an item
parameter vector of: \(\omega, \xi_1, \xi_2, \alpha_1, \tau_1, \alpha_2, \tau_2\).
Note that the GPC-MP reduces to the LMP when the number of categories is 2, and the GPC-MP reduces to the generalized partial credit model when the order of the polynomial is 1 (i.e., q=0).
Falk, C. F., & Cai, L. (2016). Maximum marginal likelihood estimation of a monotonic polynomial generalized partial credit model with applications to multiple group analysis. Psychometrika, 81, 434-460. tools:::Rd_expr_doi("10.1007/s11336-014-9428-7")
Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16, 159–176.
Other response model:
rpf.drm(),
rpf.grmp(),
rpf.grm(),
rpf.lmp(),
rpf.mcm(),
rpf.nrm()
spec <- rpf.gpcmp(5,2) # 5-category, 3rd order polynomial
theta<-seq(-3,3,.1)
p<-rpf.prob(spec, c(1.02,3.48,2.5,-.25,-1.64,.89,-8.7,-.74,-8.99),theta)
Run the code above in your browser using DataLab